Group #12014018 — subgroups

Subgroups — #12014018

Order 1280.

354 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10599854	C₂	1	4	yes	maximal, complemented
2	#10598858	C₂	1	1	yes	maximal, characteristic, complemented
3	#10598855	C₂	1	1	yes	maximal, characteristic
4	C₅×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#3932	—	2	8	no	—
6	#64988	C₂²	1	2	yes	—
7	#64984	—	2	2	no	—
8	#64984	C₂²	1	2	yes	—
9	#64971	—	2	2	no	—
10	C₄×C₅×C₁₆	C₂²	1	2	yes	—
11	C₅×C₈²	C₂²	1	1	yes	characteristic
12	#12517	—	5	1	no	maximal, Sylow
13	C₂×Dic₂₀	—	2	4	no	—
14	C₂×D₄₀	—	2	4	no	—
15	C₂×D₄₀	—	4	4	no	—
16	#4020	—	2	2	no	—
17	#4015	—	2	2	no	—
18	C₂×C₅×C₁₆	D₄	1	4	yes	—
19	C₄×C₅×C₈	D₄	1	2	yes	—
20	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
21	#3950	—	4	4	no	—
22	#3932	—	4	16	no	—
23	#2248	—	5	4	no	—
24	#1713	—	5	1	no	—
25	#1712	—	5	1	no	—
26	C₈×C₁₆	D₅	1	1	yes	characteristic, complemented
27	C₂×D₂₀	—	4	2	no	—
28	C₂×D₂₀	—	4	4	no	—
29	C₂×Dic₁₀	—	4	2	no	—
30	C₂×C₅×C₈	D₈	1	4	yes	—
31	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
32	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
33	C₄⋊Dic₅	—	4	4	no	—
34	Dic₂₀	—	4	8	no	—
35	D₄₀	—	4	8	no	—
36	D₄₀	—	8	8	no	—
37	C₅×C₁₆	D₈	1	8	yes	complemented
38	C₂×SD₃₂	—	10	8	no	—
39	#500	—	10	2	no	—
40	#494	—	5	2	no	—
41	#493	—	5	2	no	—
42	#493	—	10	2	no	—
43	C₄×C₁₆	C₂×D₅	1	2	yes	—
44	C₈²	C₂×D₅	1	1	yes	characteristic
45	C₂²×D₅	—	8	2	no	—
46	C₂×C₄×C₅	#127	1	1	yes	characteristic
47	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
48	C₂×Dic₅	—	8	2	no	—
49	D₂₀	—	8	4	no	—
50	D₂₀	—	8	8	no	—
51	Dic₁₀	—	8	4	no	—
52	C₅×C₈	SD₃₂	1	8	yes	complemented
53	C₅×C₈	C₂×D₈	1	4	yes	—
54	C₂×Q₁₆	—	10	4	no	—
55	C₂×D₈	—	10	4	no	—
56	C₂×D₈	—	20	4	no	—
57	C₄⋊Q₈	—	10	2	no	—
58	#127	—	10	2	no	—
59	SD₃₂	—	20	16	no	—
60	C₂×C₁₆	D₂₀	1	4	yes	—
61	#107	—	20	4	no	—
62	C₄×C₈	D₂₀	1	2	yes	—
63	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
64	C₂²×C₅	#493	1	1	yes	characteristic
65	C₂×D₅	—	16	2	no	—
66	C₂×D₅	—	16	4	no	—
67	C₄×C₅	#493	1	2	yes	—
68	C₄×C₅	C₂×SD₃₂	1	4	yes	—
69	Dic₅	—	16	2	no	—
70	C₂×Q₈	—	20	2	no	—
71	C₂×D₄	—	20	2	no	—
72	C₂×D₄	—	20	4	no	—
73	Q₁₆	—	20	8	no	—
74	D₈	—	20	8	no	—
75	D₈	—	40	8	no	—
76	C₂×C₈	D₄₀	1	4	yes	—
77	C₂×C₈	C₂×D₂₀	1	2	yes	—
78	C₄⋊C₄	—	20	4	no	—
79	C₄²	C₂×D₂₀	1	1	yes	characteristic
80	C₁₆	D₄₀	1	8	yes	complemented
81	C₂×C₅	#1712	1	1	yes	characteristic
82	C₂×C₅	#2248	1	2	yes	—
83	D₅	—	32	2	no	—
84	C₂³	—	40	2	no	—
85	Q₈	—	40	4	no	—
86	D₄	—	40	8	no	—
87	D₄	—	40	4	no	—
88	C₂×C₄	—	40	2	no	—
89	C₂×C₄	#4020	1	1	yes	characteristic
90	C₂×C₄	C₂×D₄₀	1	2	yes	—
91	C₈	#3932	1	8	yes	complemented
92	C₈	C₂×D₄₀	1	4	yes	—
93	C₅	#12517	1	1	yes	characteristic, complemented, Sylow
94	C₂²	—	80	4	no	—
95	C₂²	—	80	2	no	—
96	C₂²	#64984	1	1	yes	characteristic
97	C₄	—	80	2	no	—
98	C₄	#64984	1	2	yes	—
99	C₄	C₂×#3932	1	4	yes	—
100	C₂	—	160	2	no	—
101	C₂	#10598855	1	1	yes	characteristic
102	C₂	#10599854	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10599854

C₂

yes

maximal, complemented

#10598858

C₂

yes

maximal, characteristic, complemented

#10598855

C₂

yes

maximal, characteristic

C₅×C₈×C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#3932

—

#64988

C₂²

yes

—

#64984

—

#64984

C₂²

yes

—

#64971

—

C₄×C₅×C₁₆

C₂²

yes

—

C₅×C₈²

C₂²

yes

characteristic

#12517

—

maximal, Sylow

C₂×Dic₂₀

—

C₂×D₄₀

—

C₂×D₄₀

—

#4020

—

#4015

—

C₂×C₅×C₁₆

D₄

yes

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

C₂³

yes

—

—

—

—

—

C₈×C₁₆

D₅

yes

characteristic, complemented

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

Dic₂₀

—

D₄₀

—

D₄₀

—

C₅×C₁₆

D₈

yes

complemented

C₂×SD₃₂

—

—

—

—

—

C₄×C₁₆

C₂×D₅

yes

—

C₈²

C₂×D₅

yes

characteristic

C₂²×D₅

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

Dic₁₀

—

C₅×C₈

SD₃₂

yes

complemented

C₅×C₈

C₂×D₈

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

SD₃₂

—

C₂×C₁₆

D₂₀

yes

—

#107

—

C₄×C₈

D₂₀

yes

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

#493

yes

—

C₄×C₅

C₂×SD₃₂

yes

—

Dic₅

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

D₈

—

C₂×C₈

D₄₀

yes

—

C₂×C₈

C₂×D₂₀

yes

—

C₄⋊C₄

—

C₄²

C₂×D₂₀

yes

characteristic

C₁₆

D₄₀

yes

C₂×C₅

yes

C₂×C₅

yes

—

D₅

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#4020

yes

characteristic

C₂×C₄

C₂×D₄₀

yes

—

C₈

#3932

yes

complemented

C₈

C₂×D₄₀

yes

—

C₅

#12517

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

#64984

yes

characteristic

C₄

—

C₄

#64984

yes

—

C₄

C₂×#3932

yes

—

100

C₂

—

160

—

101

C₂

#10598855

yes

characteristic

102

C₂

#10599854

yes

—

	subgroup	G/H
center	C₂²	#64984
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×C₈×C₁₆	C₂
socle	C₂²×C₅	#493
prime radical	#10598855	C₂
hypercenter	C₈×C₁₆	D₅
hypocenter	C₅	#12517
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#64984

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

Fitting

C₅×C₈×C₁₆

C₂

C₂²×C₅

C₂

C₈×C₁₆

D₅

hypocenter

C₅

#12517

LCS

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #10599854	C₂	semidirect
2. #10598858	C₂	semidirect
4. C₅×C₈×C₁₆	C₂	semidirect
26. C₈×C₁₆	D₅	semidirect
37. C₅×C₁₆	D₈	semidirect
52. C₅×C₈	SD₃₂	semidirect
80. C₁₆	D₄₀	semidirect
91. C₈	#3932	semidirect
93. C₅	#12517	semidirect

H ≅ G/N

type

1. #10599854

C₂

semidirect

2. #10598858

C₂

semidirect

4. C₅×C₈×C₁₆

C₂

semidirect

26. C₈×C₁₆

D₅

semidirect

37. C₅×C₁₆

D₈

semidirect

52. C₅×C₈

SD₃₂

semidirect

80. C₁₆

D₄₀

semidirect

91. C₈

#3932

semidirect

93. C₅

#12517

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#10599854	2
#10598858	2
#10598855	2
C₅×C₈×C₁₆	2
#12517	5

subgroup

index

C₅×C₈×C₁₆

Subgroups — #12014018

Conjugacy classes of subgroups

Complements

Maximal subgroups